8 Grade
Common Core
Math
th
Booklet 4
Functions Part 1
One of the Main Idea of Functions:
Define, evaluate, and compare functions
What are functions?
Functions are like machines. You give them an input and they give you an
output. A function is essentially a rule that is assigned to an input to receive
an output.
Functions are written like this f(x) = 4x where f is the function name, (x) is
the input value and 4x represents the rule to x. Any rule to x can be used.
The rule is what the function does to x.
Examples of Functions:
f(x) = 6x - 5
g(x) = 2x + 3
f(x) = x / 2
Notice in the above examples that the letter g was used as the name of the
function. Other variables can be used to express a function name, but f is
the most common.
8th Grade Common Core Math Standards:
Standard 8.F.A.1: Understand that a function is a rule that assigns to each
input exactly one output. The graph of a function is the set of ordered pairs
consisting of an input and the corresponding output.
What the student learns: In a function, each input will only have ONE
output. A graph of a function has ordered pairs like this (input, output)
where the input corresponds to the x coordinate and the output
corresponds to the y coordinate on a graph.
Standard examples:
In a function, each input has one output because the rule assigned to the
number can only lead to one output.
The equation of this table is
f(x) = 6x +2
Input (x)
Rule (6x + 2)
Output (y)
1
6*1 + 2
8
3
6*3 + 2
20
4
6*4 + 2
26
3
6*3 + 2
20
Make the graph of this function: f(x) = 3x + 1
Input (x)
Rule
(3x + 1)
Output (y)
0
3*0 + 1
1
2
3*2 + 1
7
5
3*5 + 1
16
Coordinate Points (input, output) are:
(0,1)
(2,7)
(5,16)
Answer:
17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Standard 8.F.A.2: Compare properties of two functions each represented
in a different way (algebraically, graphically, numerically in tables, or by
verbal descriptions). For example, given a linear function represented by a
table of values and a linear function represented by an algebraic
expression, determine which function has the greater rate of change.
What the student learns: Students must be able to compare two or more
functions that are represented in a different form. For example, they should
be able to compare tables, graphs, and equations of functions and find out
which has a larger rate of change.
Standard example: Which has a greater rate of change?
!
f(x) = ! +2
Input (x)
Rule
0
!
!
!
1
!
!
2
!
!
6
!
!
+ 2
!
Output (y)
+ 2
2
+ 2
2.5
+ 2
3
+ 2
5
OR the slope of the blue line in the graph below.
8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 Answer: Remember, the slope of the blue line in the graph above is the
!!!"#$ !" ! (!"#$)
To compare the rate of change between the coordinates
!!!"#$ !" ! (!"#)
!
determined by the function f(x) = = ! +2 with the coordinates on the graph
of the blue line, let’s choose the two points (0,2) and (2,3) from the table
above. If we find the slope of the points from the table, we would subtract
the y value of the first coordinate from the y value of the second coordinate
and divide that by the x value of the second coordinate minus the x value of
the first coordinate. It is written like this (y2 - y1) / (x2 - x1) When we plug in
!!!
!
the values, we get:
or
!!!
!
!
The slope, or rate of change for the table is !
Now we need to find the rate of change, or slope, for the blue line on the graph.
We choose two coordinate points: (0,0) and (2,4)
Using the formula above of (y2 - y1) / (x2 - x1), we find the slope of the blue line
!!!
!
is
which is or 2.
!!!
!
!
The graph of the blue line has a greater rate of change (slope) because 2 > !
[The table changes 1 y per 2 x and the graph changes 2 y per 1 x]
Standard 8.F.A.3: Interpret the equation y = mx + b as defining a linear
function, whose graph is a straight line; give examples of functions that are
not linear. For example, the function A = s2 giving the area of a square as a
function of its side length is not linear because its graph contains the points
(1,1), (2,4) and (3,9), which are not on a straight line.
What the student learns: Know about linear functions defined by the
equation y = mx + b. Also learn about functions that are non-linear. Such as
the graph of y = x2
Standard example:
Is this function linear or non-linear? A linear function would have a graph of
a straight line, while a non-linear function would not have a graph in a
straight line.
State the equation (rule) of this function:
Input (x)
1
2
3
4
5
Output (y)
12
17
22
27
32
Answer: The input rises by one each time, while the output rises by five
each time.
Since everything is constant, this function is linear.
The equation of a linear line is y = mx+b
We know so far that y = 5x because the output changes by 5 every time the
input increases by 1.
Since we see that for 1, the output is 12 and for 3, the output is 22, we
know a value will be added to the 5x
If we multiply 1 by 5 we get 5. 12 - 5 is 7 so for 1, 5x + 7 gives you 12
If we multiply 3 by 5 we get 15. 22 - 15 = 7. For 3, 5x + 7 gives you 22
The equation of this table is y = 5x + 7
Are these equations linear or non-linear?
Keep in mind that anytime an equation of a line has an exponent attached
to the x value that is 1, it will be a linear equation. If the exponent of x is
greater or less than 1, it will be a non-linear equation.
Answer: Linear or non-liner? y = x3
This is nonlinear because the exponent of x is more than 1.
y = 2x + 6 is linear because the exponent of x is 1 (If you don’t see an
exponent, the exponent is 1.)
The purple line is a graph of a linear equation.
The blue line is a graph of a non-linear equation.
WHY THIS IS IMPORTANT
Functions are used all the time in the real world and you may not even
know you are using them.
Vending Machines use functions to operate. You put in a dollar and press
the combo of what you want, for example, E7 to get a pack of Skittles. After
you press the button, the machine gives you the Skittles. E7 on the
machine only has one possible output.
In real life, if you were to get two job offers, you could use functions to
figure out which job pays better. For example, Job 1 pays $15 per hour.
Job 2 pays $12 per hour plus a $100 bonus check at the beginning of every
week. To figure out which job pays more, you would express them as
functions and could graph them to see which job pays more money and
how many hours you would have to work for one job to pay off and give you
the better rate.